I don’t think so.
Here is Blue Planet II 4K HDR.
This one is from the Sony X700. When I push the “Info” button on the Projector, it says “HDR10”.
This one is from Plex. You can see that the Resolution is also 4K, but no “HDR” even though the file is HDR.
Can you push the Info button or equivalent to see what the Projector or TV is actually showing?